Powerset of Subset is Closed under Intersection

Theorem

Let $S$ be a set.

Let $T \subseteq S$ be a subset of $S$.

Let $\powerset S$ denote the power set of $S$.

Then $\powerset T$ is a closed subset of $\powerset S$ under set intersection:

$\forall A, B \in \powerset T: A \cap B \in \powerset T$

Proof

A direct application of Power Set is Closed under Intersection.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.3$